ON GENERALIZED EPI-PROJECTIVE MODULES

A module M is said to be generalized N-projective (or N-dual ojective) if, for any epimorphism g : N &#8594; X and any homomorphism f : M &#8594; X, there exist decompositions M = M1 &#8853; M2, N = N1 &#8853; N2, a homomorphism h1 : M1 &#8594; N1 and an epimorphism h2 : N2 &#8594; M2 such that g &#9702; h1 = f|M1 and f &#9702; h2 = g|N2 . This relative projectivity is very useful for the study on direct sums of lifting modules (cf. [5], [7]). In the definition, it should be noted that we may often consider the case when f to be an epimorphism. By this reason, in this paper we define relative (strongly) generalized epi-projective modules and show several results on this generalized epi-projectivity. We apply our results to the known problem when finite direct sums M1&#8853;· · ·&#8853;Mn of lifting modules Mi (i = 1, · · · , n) is lifting.</p

ON MONO-INJECTIVE MODULES AND MONO-OJECTIVE MODULES

In [5] and [6], we have introduced a couple of relative generalized epi-projectivities and given several properties of these projectivities. In this paper, we consider relative generalized injectivities that are dual to these relative projectivities and apply them to the study of direct sums of extending modules. Firstly we prove that for an extending module N, a module M is N-injective if and only if M is mono-Ninjective and essentially N-injective. Then we define a mono-ojectivity that plays an important role in the study of direct sums of extending modules. The structure of (mono-)ojectivity is complicated and hence it is difficult to determine whether these injectivities are inherited by finite direct sums and direct summands even in the case where each module is quasi-continuous. Finally we give several characterizations of these injectivities and find necessary and sufficient conditions for the direct sums of extending modules to be extending